The concept of Leibniz algebras was introduced by Loday [1]}, [2]} in the study of the algebraic \(K\) -theory. Relative Rota-Baxter
operators on Leibniz algebras were studied in [3]}, which is the main ingredient in the study of the twisting theory and the bialgebra theory for Leibniz algebras [4]}. Recently, the author has considered weighted relative Rota-Baxter operators on Leibniz algebras in [5]}. Our aim in this paper is to consider crossed homomorphisms between Leibniz algebras using Leibniz representation introduced by [5]}, More precisely, we construct a differential graded Lie algebra whose Maurer-Cartan elements
are given by crossed homomorphisms on Leibniz algebras. This allows us to define cohomology
for a crossed homomorphism. Finally, we study linear deformations, formal deformations and extendibility of finite order deformations of a crossed homomorphism in terms of the cohomology theory.
